Vector beams encoded by diverse orthogonal polarization states and their generation based on metasurfaces

Vector beams with spatially varying polarization states have wide application prospects. Convenient and feasible generation of compact vector beams become important for the applications in integrated optics and optical communication. Based on the superposition of orthogonal polarization vortex beams and the metasurface design, this paper studies systematically vector beams with hybrid states of polarization in diverse orthogonal polarization encoding. The spatial polarization of vector beam can be controlled through choosing the polarization types and adjusting the contributions of two orthogonal polarized beams. The generated vector beams based on metasurfaces verify the output of diversity polarization states in the transverse plane. The high polarization sensitivity of vector beam is beneficial to the polarization multiplexing and the utilization of optical metasurface realizes the polarization manipulation in nanometer scale. This work lays the foundation for the construction of complex vector beams and it is helpful for expanding the applications of vector beams.


Introduction
Vector beam points to the light beam with spatially varying polarization state, like our familiar radial and azimuthal vector beams, where the orientation of linear polarization state in light field changes with the angular position [1,2]. Because of the special polarization distribution during the propagation and the interaction with the matter, vector beam shows some unique characteristics such as tight focusing [3], gradient force [4] and high polarization sensitivity. Its optical performance obviously surpasses that of scalar beam with the uniform polarization state in some particular occasions. Therefore, vector beams have attracted much attentions and they have widely been applied in optical tweezers [5], optical sensing [6], material processing [7], microscopy imaging [8] and quantum memories [9].
The output of the common laser is scalar beam and the polarization conversion needs to be implemented to obtain vector beam. Several techniques have been proposed to realize the conversion from the uniform polarization to nonuniform polarization. In general, the generation of vector beam has two ways. One is to realize the polarization control with the aid of the local response of optical element. Sub-wavelength grating [10], spiral phase plate [11], q-plate [12,13] and metasurface [14][15][16] have been also advised to manipulate the polarization state of local light field and directly generate vector beam. Vector beams consisting of spatially varying linear polarization states are formed using this techniques. And our group also generated radial and azimuthal vector beams using the local polarization conversion of nanounits [14][15][16]. The spatially varying linear polarization states can be easily generated by manipulating the local response of nanounits. However, the complex polarization distribution consisting of hybrid states of polarization is difficult to realize in terms of the identifying nanounits and simple design.
The other way is to obtain spatially varying polarization through superposing two orthogonal polarization beams. Cylindrical lens [17], spatial light modulator [18][19][20][21] and interference system [22][23][24] have been utilized to superpose two orthogonal linearly or circularly polarized beams and generate vector beam. Till now, cylindrical vector light and the extended Poincaré beams are obtained using these methods [25][26][27]. And our recent work performs the related study and obtains two kinds of vector beams with different polarization singularities through superposing the horizontal and vertical linearly polarized beams and left-and right-handed circularly polarized beams using optical metasurfaces [28]. The vector beams generated by this method have the spatially varying polarization type, polarization orientation and polarization ellipticity. Obviously, this method can manipulate more polarization degrees of freedom of the light field with respect to the former local field control method, and the polarization type, and polarization ellipticity may be adjusted except for the orientation of polarization.
As we know, the superposition of different orthogonal modes may generate different vector beams. Orthogonal polarization states have diverse forms, like the orthogonal linearly polarized states rotated in Cartesian base and two any orthogonal elliptically polarized states. However, the polarization characteristics of vector beams encoded by different orthogonal polarization states are not clear. How to generate this kind of vector beams in nanoscale is also an unsettled issue. This paper proposes the study about the vector beams with hybrid states of polarization (VBHSP) superposed by diverse orthogonal polarized states. The universal expression of VBHSP encoded by arbitrary orthogonal polarization bases is provided, which covers the traditional vector beams encoded by Cartesian base and circular polarization base. In view of the powerful light manipulation ability of metasurfaces [28,29] and the research foundation in the generation of VBHSP based on metasurfaces [14][15][16][30][31][32], this paper realizes the generation of VBHSP generation using optical metasurface to superpose any orthogonal polarized vortex beams. The rational to generate vector beams is exhibited and several templates are utilized to illustrate the reliability of theoretical expression and the feasibility of metasurface design. The detailed content is organized as follow. Section 2 gives the general expression of VBHSP and reveals the dependence relation of the polarization distribution on two orthogonal beams. Section 3 provides the polarization distributions of the proposed vector beams and intuitively exhibits the variation of vector light field with the elementary beams. The metasurfaces consisting of cross nanoholes equivalent to half wave plates are designed to generate VBHSP with diverse polarization states and section 4 shows the design principle. Section 5 gives the simulation and experimental validations for the generation of vector light. The discussions in section 6 about the vector beams superposed by other orthogonal polarization states enrich the concept of vector beams.
The main contributions of this paper include two aspects. One is to provide the theoretical foundation for the generation of VBHSP. The proposed universal expression can be available for the generation of diversified vector beams with spatially varying polarization type, polarization orientation and ellipticity. The other is to propose the generation method of VBHSP based on metasurface. The utilization of metasurface enables the miniaturized and integrated vector beam generators. As far as we know, it is the first time to generate and characterize the VBHSP carrying plentiful polarization information, which forms with the help of any orthogonal polarized states. We hope the generated VBHSP based on metasurface can be applied in wider fields including particle micro-manipulation and optical integration.

Theoretical model
Here, we study a kind of VBHSP superposed any two orthogonal polarization beams and the polarization distributions of this kind of vector beams at any spatial position only depend on the azimuthal angle of φ. The expression of this kind of vector beams can be written in the following unified form where two Jones vectors represent two orthogonal polarization states. The angle β is responsible for the orientation of particular polarization and the angle γ is responsible for the particular polarization type. While γ equals to 0 and β takes a certain value, two Jones vectors denote orthogonal linearly polarized states. While γ equals to π/2 and β takes a certain value, two Jones vectors are orthogonal elliptically polarized states. It can be seen that β and γ determine the polarization types of Jones vectors. Two exponential terms indicate two orthogonal polarization beams carry spiral phases and they are optical vortices with the integers of m and n being the topological charges of vortices. The angle of θ determines the amplitude ratio of two orthogonal polarization beams. Among the above equation, the angle of θ determines the contribution of two orthogonal beams and it changes within the region of [0, π/2]. The angle of ϕ take the value between 0 and π, and the angles β and γ are limited in the region of [0, π/2]. Obviously, the polarization distribution of vector beam depends on six parameters including four angles of θ, ϕ, β and γ, and two integers of m and n. Next, vector beams are divided into two cases to analyze in detail their polarization characteristics. The first one is the case of γ = 0 and it is the orthogonal linear polarization encoding. The second one is the case of γ = π/2 and it is the orthogonal elliptical polarization encoding.

Orthogonal linear polarization encoding
The orthogonal linear polarization encoding points to the case that two vortex beams are chosen as the orthogonal linearly polarized states, and the expression of vector beam can be deduced from equation (1), As β = 0, two Jones vectors denote (1, 0) and (0, −1) and they are the unit vectors along the x and −y axis directions. As β = π/4, two Jones vectors denote 2 −0.5 (1, 1) and 2 −0.5 (1, −1) and they are the unit vectors along the diagonal and anti-diagonal directions. And As β = π/2, two Jones vectors denote (0, 1) and (1, 0) and they are the unit vectors along the y and x axis directions. In order to directly reveal the dependence relationship of vector beam on two orthogonal polarization beams, we rewrite equation (2) in the matrix multiplication form where p = (m + n)/2 and q = (m − n)/2. The exponential term at the front denotes the generated vector beam brings a spiral phase. The Jones vector at the right means this is one elliptically polarized state with the polarization ellipticity of vector field depending on qφ + ϕ. When qφ + ϕ = 0, π/2, π or 3π/2, the Jones vector denotes the linear polarization. When qφ + ϕ = π/4, 3π/4, 5π/4 or 7π/4, the Jones vector denotes the circular polarization. At other azimuthal positions, the Jones vector represents the elliptical polarization. In view of the spatially varying polarization type of vector field, we call this VBHSP as the elliptically polarized vector beam. The Jones matrix in equation (3) depends on the value of β, namely, the choice of the orientations of two orthogonal polarization beams, and it also relates to the value of θ, namely the amplitude ratio of two orthogonal polarization beams. This means the polarization distribution of VBHSP changes with the values of β and θ. As the amplitudes of two orthogonal polarization beams are equal, namely, θ = π/4, the Jones matrix is equivalent to a rotation matrix with the rotation angle of π/4−β and the polarization direction of VBHSP changes with the angle of β.

Orthogonal elliptical polarization encoding
The orthogonal elliptical polarization encoding points to the case that two vortex beams are the cross elliptically polarized states. While the angle of γ equals to π/2, equation (1) can be written into As 0 < β < π/4, two Jones vectors denote the orthogonal polarization states with the ellipticity of the former smaller than that of the latter. As π/4 < β < π/2, two Jones vectors denote the orthogonal polarization states with the ellipticity of the former larger than that of the latter. Certainly, as β = 0 or π/2, two Jones vectors return to the linear polarization states with the phase difference of π/2. Moreover, as β = π/4, two Jones vectors are 2 −0.5 (1, j) and 2 −0.5 (1, −j) and they are the unit vectors of the left-and right-handed circularly polarized states. The vector beam superposed by two cross circularly polarized beams can be also expressed into the matrix multiplication form The Jones vector in above equation denotes one elliptically polarized state with the polarization ellipticity depending on the angle of θ. Obviously, as the angle of θ takes a certain value, the polarization type of vector beam at any position is ascertained. As θ = π/4, the vector field at any position is linear polarization and we usually call this vector beam as the linearly polarized vector beam. The Jones matrix is equivalent to a rotation matrix with the rotation angle of qφ + ϕ. And the exponential term at the front denotes the generated vector beam also brings a spiral phase.

Polarization analysis
In order to intuitively exhibit the polarization characteristics of VBHSP superposed by orthogonal vortex beams with different polarization types, we depict the polarization distributions of vector beams. Figure 1 gives the polarization distributions of vector beams superposed by the orthogonal linearly polarized beams with the orientations, amplitude ratio, initial angle and topological charges of two orthogonal vortex beams taking different values. Figure 1 For the polarization distributions in figure 1(a), the parameters of (m, n, θ, ϕ) from left to right correspond to (1, −1, π/4, 0), (2, −2, π/4, 0), (2, −2, π/8, 0) and (2, −2, π/4, π/4). Comparing two patterns at the left, one can see that the values of m and n influence the polarization ellipticity and polarization direction of vector field. The pattern with m = 1 and n = −1 is two-fold cylindrical symmetric, and the pattern with m = 2 and n = −2 is four-fold cylindrical symmetric. Comparing two patterns in the middle, one can see the change of θ only influences the polarization orientation and the ellipticity at any position, but the handedness of polarization state is unchangeable. From two patterns at the second and fourth columns, one can see that the value of ϕ influences the ellipticity of the polarization state, and in fact, the polarization state with the same ellipticity appears at the position angle of ϕ/q.
The results in figures 1(b) and (c) show the similar polarization variation rules of vector beam with m, n, θ and ϕ. Moreover, from three patterns at any column, one can see that the polarization direction of vector beam at any position rotates with increase of the angle of β, which can be clearly seen from the polarization states at the positions squared by green lines. Yet the ellipticity and the handedness of the polarization state do not change with the angle of β. Figure 2 gives the polarization distributions of vector beams encoded by the orthogonal elliptically polarized beams, where the orientations, amplitude ratio, initial angle and topological charges of two orthogonal beams also take different values. Figure 2(a) gives the results with β = π/6, figure 2(b) shows the results with π/4 and figure 2(c) gives the results with π/3, where the arrows at the left of patterns denote different orthogonal elliptical polarization states. The parameters of (m, n, θ, ϕ) for the patterns from left to right in figures 2(a)-(c) correspond to (−1, 1, π/4, 0), (−2, 2, π/4, 0), (−2, 2, π/8, 0) and (−2, 2, π/4, π/4).
The patterns on two left columns of figure 2(a) show that the values of m and n influence the polarization ellipticity and polarization direction of vector field. Like the cases in figure 1(a), the pattern with m = −1 and n = 1 is two-fold cylindrical symmetric, and the pattern with m = −2 and n = 2 is four-fold cylindrical symmetric. The patterns on two middle columns of figure 2(a) show that the angle of θ only influences the polarization ellipticity but does not influence the orientation of the polarization state at any position. Two patterns on the second and fourth columns of figure 2(a) show that the polarization state with the same ellipticity rotates at the angle of ϕ/q. The results of figures 2(b) and (c) take on the similar variation rules. Comparing the patterns on the same column of figures 2(a)-(c), one can see for the given parameters of (m, n, θ, ϕ), the ellipticity of the polarization state change with the increasing of β, and yet the polarization direction has no change. Like the polarization states at the positions squared by green lines, the polarization type changes from the right-handed elliptical polarization, the linear polarization to the left-handed elliptical polarization. Specially, the results in figure 2(b) show the polarization type of light field at any position is identical, as the theory predicted. The polarization type of vector beam may be identical linear or elliptical polarization, and it is completely different from the other cases.

Metasurface design
Here, we generate the VBHSP superposed by any orthogonal polarization vortex beams based on metasurface consisting of cross nanoholes with two-fold rotation symmetry. For this anisotropic nanohole with its fast axis rotating the angle of α, its transmission can be expressed by the following matrix, where t o and t e denote the transmission amplitudes of the nanohole along the fast and slow axes, and δ is the phase difference of transmission fields along two directions. These parameters depend on the size and shape of nanohole. Under the circularly polarized light illumination, the transmission field of nanohole can be expressed by The first term has the same polarization as the incident beam, and the second term is the cross circular polarization. The amplitudes of two parts are different, and when the nanohole can be equivalent to a half wave plate, namely, t o = t e and δ = π, the first term disappears and the cross polarization conversion efficiency reaches the maximum. In practice, the equivalent half wave plate can be obtained through optimizing the parameters of nanohole, like the cross nanoholes proposed by our group [33,34]. Moreover, we notice that an additional phase term of e ±j2α is carried by the cross circular polarization. This is the reason that the rotated nanohole can introduce an additional phase. As the orientation angle of nanohole with respect to the long side at the coordinates of (r, φ) satisfies   Figure 3(b) shows the structure of metasurface combining two suits of nanoholes and it can generate the vector beam superposed by two cross circularly polarized beams, where two suits of nanoholes are denoted by different colors.
As we know, one linear polarization state, like the cases of (cosβ, sinβ) and (sinβ, −cosβ), can be taken as the superposition of two circular polarization states, Therefore, the linear polarization encoding vector beam can be taken as the superposition of four circularly polarized beams carrying the extra phases, like the ones appearing in equation (9). Thus, four suits of nanoholes need to be integrated to construct the metasurface and figure 3(c) shows the schematic diagram of this metasurface, where four suits of nanoholes are denoted by different colors. This metasurface can generate the vector beam superposed by any two orthogonal linearly polarized beams. Furthermore, it needs to be pointed out that any two orthogonal elliptically polarized beams can be also expressed by the circularly polarized beams, Therefore, the general elliptical polarization encoding vector beam can be taken as the superposition of eight circularly polarized beams carrying the different extra phases. Thus, eight suits of nanoholes need to be integrated to construct the metasurface and this metasurface can generate the vector beam superposed by any two orthogonal elliptically polarized beams.

Polarization performance verification
We first optimize the parameters of the nanohole etched in the silver film so that the nanohole can be taken as a half wave plate for the illumination wavelength of 633 nm. The thickness of silver film takes 220 nm. The sizes of cross nanohole are l 1 = 600 nm, w 1 = 150 nm, l 2 = 180 nm and w 2 = 220 nm. Then the optimized cross nanoholes are arranged in concentric circles with the separation of 613 nm and the vector beam is generated by the designed metasurface. All these works are performed with the help of the finite-difference time-domain technique [35]. In simulation calculations, the perfectly matched layers are used as the boundary to prevent non-physical scattering, and the minimum mesh step is set at 2 nm to ensure the calculation precision. The dielectric constant of silver is taken the value given by Palik [36]. The detection plane is set at 6 µm above the metasurface. Figure 4 gives the detected polarization distributions for vector beams superposed by two orthogonal linearly polarized beams with θ = π/4, ϕ = 0 (left) and π/4 (right), and β = 0, π/4 and β = π/2 (from top to down). The arrows inserted at the left of the patterns denote the polarization states of two linearly polarized states. Figure 4 The generated vector beams have diverse polarization states. Figure 5 gives the detected polarization distributions for vector beams superposed by two orthogonal elliptically polarized beams with θ = π/4, ϕ = 0 (left) and π/4 (right), and β = π/6, π/4 and β = π/3 (from top to down). The arrows inserted at the left of the patterns denote the polarization states of two elliptically polarized states. Figure 5(a) gives the results for vector beams with the polarization order taking 1 (m = −1 and n = 1) and figure 5(b) gives the results for vector beams with the polarization order taking 2 (m = −2 and n = 2). One can see that the polarization distributions in figure 5 are almost consistent to the results in figure 2, and minor imperfections do not substantially affect the detected outcome.
These simulation processes may be replaced by the practical experiment. The manufacture of sample includes the deposition of silver film and the fabrication of nanoholes. The silver film with 220 nm thickness can be deposited on the glass substrate using the magnetron sputtering deposition method and the nanoholes can be fabricated using the focused ion-beam etching technique. The laser beam with the incident polarization state adjusted by the combination of one quarter-wave plate and one polarizer impinges upon the metasurface sample from the glass substrate. The magnified transmission field by the microscopic objective may be captured using one CCD camera. Figure 6(a) shows the experimental setup, where QWP denotes the quarter-wave plate, P1 and P2 are the polarizers, M1 and M2 are the mirrors and S is the metasurface sample. Figure 6(b) gives the scanning electron microscopy (SEM) images of one sample, which can generate the vector beam of m = 2, n = −2, θ = π/4 and ϕ = 0 with linear polarization encoding. Figure 6(c) shows the measured intensity distributions without and with polarization analyzer, where the inserted yellow arrows denote the transmission direction of polarizer.
From figure 6(c), one can see that the intensity distribution of vector beam without polarization analyzer takes on the doughnut shape and the polarization distribution is four-fold rational symmetric. The intensity distribution of vector beam with the transmission direction of polarizer is along the horizontal axis, four bright spots appear at the left and right, and top and down positions. And the intensity distribution of vector beam with the transmission direction of polarizer is along the vertical axis, four bright spots appear at the diagonal and anti-diagonal positions. These distributions tally with the polarization distribution rule. For convenience comparison, figure 6(d) also give the simulated intensity distributions with and without polarization analyzer. It can be clearly seen that the experimental results are consistent with the simulation results.

Symmetry of polarization distribution
From the results in figures 1 and 2, one can see that the polarization distributions are cylindrical symmetric. It is easy to find that all these results correspond to the cases with q = (m − n)/2 taking the integer. As q takes the fraction, the polarization distribution of vector beam may be different, and we also simulate the polarization distributions of the vector beams superposed by linearly and circularly polarized beams with q  where θ = π/4, ϕ =0 (left) and π/4 (right) and β = π/6, π/4 and π/3 (from top to down). taking different fraction. Figure 7(a) shows the results of the vector beams superposed by orthogonal linearly polarized beams with q = 1/2 (m = 2 and n = 1 for the left) and 3/2 (m = 2 and n = −1 for the right), where the angle of θ takes π/4 and the angle of β from top to down take 0, π/4 and π/2. Figure 7(b) shows the results of the vector beams superposed by orthogonal circularly polarized beams with q = −1/2 (m = 1 and n = 2 for the left) and −3/2 (m = −1 and n = 2 for the right), where the angle of θ takes π/4 and the angle of β from top to down take π/6, π/4 and π/3.
Obviously, the polarization distribution of vector beam with q taking the fraction has no cylindrical symmetry and the polarization distribution of vector beam changes with the value of q. For the vector beams in linear polarization encoding, the polarization direction at any position changes with the orthogonal polarization states, but the ellipticity does not change. For the vector beams in circular polarization encoding, the ellipticity at any position changes with the orthogonal polarization states, but the polarization direction does not change. These distribution rules are the same as those of vector beams with q taking the integer.

Tilted elliptical polarization encoding
In above analysis, the angle of γ takes 0 or π/2 and the angle of β takes different value. As the angle of β takes a certain value and γ varies within the region of (0, π/2), the directions of two orthogonal elliptical  Polarization distributions of vector beams superposed by (a) the linearly polarized beams with q = 1/2 (left) and 3/2 (right) and β = 0, π/4 and π/2 (b) circularly polarized beams with q = −1/2 (left) and −3/2 (right) and β = π/6, π/4 and π/3. polarization states tilt at the angle of β and the ellipticities change with the value of γ. In order to intuitively show the polarization distribution rules of vector beams superposed by tilted elliptically polarized beams, we also simulate the polarization distributions of the corresponding vector beams with β = π/4 and γ taking π/6, π/4 and π/3. Figure 8  direction and handedness of polarization state do not change. Comparing the results of figures 8 and 2, one can see the polarization distribution rules are similar except the ellipticity. Certainly, as the angle of γ decreases to 0, the polarization distribution turns into the case of figure 1(b), and as the angle of γ increases to π/2, the polarization distribution changes into the case of figure 2(b).

Radical and azimuthal polarization encoding
As we know, the radial and azimuthal polarization states are orthogonal. Although the polarization directions of radial and azimuthal polarization states change with the spatial position, which are different from the case in the linear polarization encoding discussed in section 2, their polarization types are linear polarization. Broadly speaking, we may classify this vector beams in the case of the linear polarization encoding. The radially and azimuthally polarized beams can be also used to encode the vector beam and the expression of this vector beam can be obtained by submitting β = φ into equation (2) or (3). Figure 9 shows the polarization distributions of vector beams superposed by radially and azimuthally polarized beams with the angles of θ and ϕ and the values of m and n taking different values. Comparing the patterns in figure 9(a), one can see the polarization distributions change with m and n. As q takes an integer, the polarization distribution has cylindrical symmetry, like the cases of m = 1 and n = −1, and m = 2 and n = −2. As q takes a fraction, the polarization distribution has no cylindrical symmetry, like the cases of m = 2 and n = 1, and m = 2 and n = −1. For the patterns with the same m and n in figures 9(a) and (b), the polarization direction at any position change with the increasing of θ, yet the handedness of polarization state does not change. For the patterns with the same m and n in figures 9(b) and (c), one can see the rotation angle of the polarization state with the certain ellipticity still equals to ϕ/q.

Poincaré-like sphere illustration
Like the case of Poincaré sphere to represent the polarization state, the VBHSP superposed by any orthogonal polarization beams can be illustrated by the points on the surface of Poincaré-like sphere. Figure 10    sphere, which indicates that the polarization distribution varies with the angle of ϕ. The polarization distributions on four points of each Poincaré-like sphere are shown, where m and n take 1 and −1 for figures 10(a), (b) and (d), and they take −1 and 1 for figure 10(c). These four points correspond to two poles of Poincaré-like sphere with values of θ taking 0 and π/2, and two points on the equator of Poincaré-like sphere with θ = π/4, ϕ = 0 and π/4. The polarization distributions for other points on the surfaces of Poincaré-like spheres can be also illustrated with the corresponding angles of θ and ϕ. In fact, the vector beams superposed by any orthogonal polarization beams can be illustrated by the similar pictures.

Conclusions
In this paper, we construct a kind of VBHSP superposed by any orthogonal polarization vortex beams and provide the general expression of VBHSP. Two orthogonal polarization beams may be linear polarization, circular polarization, elliptical polarization or radial and azimuthal polarization. The superposition of different orthogonal polarization states causes different polarization distributions. The matrix expression reveals the dependence relation of the polarization distribution of vector beam on the parameters of two orthogonal polarization beams. The simulation results intuitively exhibit the polarization distribution rules of vector beams. The metasurfaces consisting of rotated cross nanoholes are designed to realize the superposition of two orthogonal linearly and two orthogonal circularly polarized beams, and the detected field distributions verify the feasibility of theoretical models of vector beams. The generation of VBHSP with diverse polarization distributions demonstrates the powerful light manipulation ability of metasurface. The theoretical work of this paper lays the foundation for VBHSP generation and the utilization of metasurface design is benefit to expanding the applications of vector beams in integrated optics.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.